# Basic Material Properties

## Stress ( \( \sigma \) ): Internal force per cross-sectional area

$$ \sigma = \frac{F}{A} $$

## Strain ( \( \epsilon \) ): Ratio of elongation to original

$$ \epsilon = \frac{\Delta L}{L} $$

## Thermal Strain ( \( \epsilon _T \) ), Thermal Stress ( \( \sigma _T \) ):

$$ \epsilon _T = \alpha \Delta L $$

$$ \sigma _T = E \alpha \Delta L $$

Where \( \alpha \) is the linear coefficient of thermal expansion.

If there are no external forces or restraints on the expansion, thermal stress is zero.

## Poisson's Ratio

Ratio of lateral strain to longitudinal strain.

$$ v = - \frac{lateral}{longitudinal} $$

## Hooke's Law - Young's Modulus

$$ E = \frac{\sigma}{\epsilon} $$

## Theory of Bending

$$ \frac{M}{I} = \frac{\sigma}{y} = \frac{E}{R} $$

Where \( M \) is the external moment acting on the beam, \( I \) is the second moment of area, \( \sigma \) is the stress of the material, \( y \) is the distance from the neutral layer, \( E \) is Young's Modulus, and \( R \) is the radius of curvature.

\( I \) for a rectangular cross section:

$$ I = \frac{bd^3}{12} $$

Where \( b \) is parallel to the neutral layer.

\( I \) for a circular cross section:

$$ I = \frac{\pi d^4}{64} $$

Where \( d \) is the diameter of the cross section.

## Parallel Axis Theorem

$$ I_{xx} = I_{gg} + Ah^2 $$

Where \( I_{gg} \) is the second moment of area passing through the centroid of body \( G \), \( I_{xx} \) is the second moment of area parallel to \( I_{gg} \) at a distance \( h \) away. \( A \) is the area of body \( G \).

## Average Shear Stress ( \( \tau \) ):

$$ \tau = \frac{shearforce}{forcearea} = \frac{P}{A} $$

Where \( P \) is the internal shear force.

Assuming equilibrium conditions, average shear stress on the middle body is

$$ \tau = \frac{\frac{P}{2}}{A} $$

## Shear Strain

$$ \gamma \approx tan \gamma $$

$$ G = \frac{\tau}{\gamma} $$

Where \( G \) is the shear modulus.

## Shear Stress in Beams

$$ \tau = \frac{VQ}{It} $$

Where \( V \) is the internal shear force, \( Q \) is the first moment of area of the half of the member's cross-section where \( t \) is measured, \( I \) is the second moment of area of the entire cross-section, \( t \) is the width of the cross-section at the point where \( \tau \) is to be determined.

## Torsion

$$ \gamma = \frac{r \theta}{l} $$

Where \( r \) is the radius of the beam, \( \theta \) is the angle of distortion in radians, \( l \) is the length of the beam.

$$ \frac{ \tau }{r} = \frac{G \theta}{l} $$

The resultant torque \( T \) for a thin-walled tube:

$$ T = 2 \pi r^2 \tau t $$

Where \( t \) is the thickness of the wall.

$$ \frac{T}{J} = \frac{ \tau }{r} = \frac{G \theta}{l} $$

Where \( T \) is the resultant torque, \( J \) is the polar second moment of area, \( \tau \) is the shear stress, \( r \) is the radius of the beam, \( G \) is the shear modulus, \( \theta \) is the angle of distortion in radians, \( l \) is the length of the beam.

The polar second moment of area for a shaft of circular cross-section is

$$ J = \frac{\pi d^4}{32} $$

For a hollow shaft:

$$ J = \frac{ \pi (d_2 ^4 - d_1 ^4 )}{32} $$

Where \( d_1 \) is the inner diameter, \( d_2 \) is the outer diameter.